Time series analysis and smoothing (QCE General Mathematics Unit 3)

What this dot point is asking

This page covers Unit 3 Topic 3: Time series analysis in the QCAA General Mathematics 2025 syllabus. The topic has two parts:

• Describing and interpreting patterns in time series data: construct and interpret time series plots, and describe them by identifying trend (long-term direction), seasonality (systematic, calendar-related movements) and irregular fluctuations (unsystematic, short-term fluctuations).
• Analysing time series data: smooth a series with a simple moving average using the mean or median for an odd number of data (including with spreadsheets), deseasonalise using seasonal indices found by the average percentage method, fit a least-squares line to model the long-term trend, and solve practical problems, including forecasting.

This page teaches the first part in depth, plus smoothing, and gives a complete overview of the second part. The seasonal index calculations are developed step by step on seasonal indices and deseasonalising. The topic appears in the internal examination (IA2), suits the problem-solving and modelling task, and appears in the external assessment.

The answer

What a time series is

A time series is a set of observations of one variable recorded at regular intervals of time: daily temperatures, monthly unemployment rates, quarterly sales, yearly rainfall. Because the values are in time order, you cannot shuffle them, and the most useful questions are about how the variable changes over time.

A time series plot graphs the variable (vertical axis) against time (horizontal axis), with consecutive points joined by straight line segments so the eye follows the movement from one period to the next. When constructing one:

• put time on the horizontal axis with equal spacing for equal intervals;
• label both axes with units, and label the time periods clearly (for example Q1 2024, Q2 2024);
• join the points in order; do not draw a line of best fit through them.

Describing a time series plot

Describe the plot by commenting on each of the three syllabus features, always in context.

Trend

The long-term direction of the series: increasing, decreasing or no clear trend. Say whether it looks roughly linear, and quote values if they help ("rising from about 40 to about 60 units over three years").

Seasonality

A pattern that repeats at the same time in each calendar cycle: a peak every summer, a trough every February, higher sales every fourth quarter. State the period (yearly, weekly) and where the peaks and troughs fall. Seasonality is systematic: you can predict when it will happen.

Irregular fluctuations

Movements with no pattern, caused by chance or one-off events (a storm, a strike, a promotion). They are unsystematic. A single extreme value caused by a one-off event is sometimes called an outlier; describe it and give the likely cause if the context suggests one.

A model description: "The number of guests shows an increasing trend over the four years. It is seasonal, peaking every December quarter and falling to a low every June quarter. There is an irregular fluctuation in the March quarter of 2022, when guest numbers fell sharply, possibly because of flooding."

Smoothing with a moving average

Irregular fluctuations can hide the trend. Smoothing replaces each value with the average of a small group of consecutive values centred on it. The syllabus uses an odd number of data (3-point, 5-point, 7-point and so on), so that each group has a middle value and each smoothed value lines up with an actual time period.

• Moving mean: the mean of the group.
• Moving median: the median of the group. It is barely affected by one extreme value, because an extreme value is never the middle of three or five.

For a $k$-point moving average ($k$ odd), $\frac{k - 1}{2}$ values are lost at each end: one at each end for 3-point, two at each end for 5-point.

Choosing the number of points

• More points gives a smoother series but loses more values at each end and responds more slowly to genuine changes in direction.
• Match the window to the season when you can. A moving mean whose length equals the length of the seasonal cycle contains exactly one of each season, so it removes the seasonal pattern completely. For daily data with a weekly pattern, a 7-point moving mean does this, and 7 is odd, so it fits the syllabus exactly.
• Use a moving median when there are outliers. One faulty reading can pull several moving means well away from the true level.

In a spreadsheet, enter the first average with a formula such as =AVERAGE(B2:B4) or =MEDIAN(B2:B4) beside the middle value (row 3), then fill down. For a 7-point average, use =AVERAGE(B2:B8) beside row 5.

Overview: seasonal indices, deseasonalising and trend lines

Moving averages reduce irregular fluctuations; seasonal indices measure and remove seasonality exactly.

Seasonal index. Compares a season with the average for its cycle. An index of 1.25 means that season is typically 25% above average; 0.80 means 20% below. The indices for one cycle add to the number of seasons (4 for quarters, 12 for months).

Average percentage method. (1) Find the average of each year. (2) Express each value as a percentage of its own year's average. (3) Average these percentages for each season across the years. (4) Write each index as a decimal and check the sum.

Deseasonalise by dividing: $\text{deseasonalised value} = \dfrac{\text{actual value}}{\text{seasonal index}}$. Reseasonalise by multiplying.

Trend line. Number the periods $t = 1, 2, 3, \ldots$ without restarting each year, fit a least-squares line to the deseasonalised values, forecast by substituting a future $t$, then multiply by that season's index.

Interpreting a trend line and judging a forecast

When a least-squares line $y = a + bt$ models a trend:

• the slope $b$ is the average change in the variable per time period (per quarter, per month), stated with units;
• the intercept $a$ is the model's value at $t = 0$, the period before the first data point, which may or may not be meaningful;
• every forecast is an extrapolation, so it assumes the trend (and seasonal pattern) continue. Short-range forecasts are more reliable than long-range ones; one-off events and changing conditions make distant forecasts unreliable.